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I was pretty sure that 切{き}った meant something was cut from something else. So ケーキから一人分{ひっとりぶん}を切{き}った would mean, "one piece was cut from the cake."

However, I came across a usage which, at least at first, doesn't seem to conform to that understanding. In my JLPT textbook, there is this graph showing the number of people in the farming industry (ignore the red line, it's just a pen mark):

One question asked which of four sentences best describes the change from the beginning to the end of the graph. This is the correct answer:

一千{いっせん}万人{まんにん}を上回{うわまわ}っていたものが、300万{まん}人{にん}を切{き}った。

My erroneous interpretation was that the sentence meant something like "There were over 10 million people, but 3 million were cut." Or, in other words, 3 million cut from 10 million leaves 7 million remaining. But the graph shows it's 3 million remaining, not 7 million, so I didn't think this was the right answer.

Instead, after checking the answers and seeing that this sentence is indeed the correct description, I'm now guessing the sentence is supposed to mean "There were over 10 million people, but it was cut down to 3 million."

However, I can't see how in this sentence 切{き}った means "cut down to", and not "cut from".

What is it that I am not seeing in this sentence, or what am I not understanding about 切{き}る?

I'm on board with the idea that 切る doesn't correlate directly with cut. The only reason I'm finding it hard to wrap my head around it is two reasons. One is that all my dictionaries don't show any meaning like "fall below". For example, Space ALC. The other reason is that in all the years I've been in Japan, I've never encountered this variant of 切る before. Again, I'm not trying to deny it means what you say, just that it's strange to me how it works. "Cut to" and "cut from" are opposing concepts, so how can the same word do both?
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QuestionerMay 18 '12 at 15:09

1

How about if you visualize it as a line on a graph "cutting" through another line as it goes down?
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MattMay 18 '12 at 20:40

@Matt Yes, I think that's how I grasp the word, it's like cutting 10 seconds into 9.9 + 0.1, cutting 10,000 yen into 9,980 yen + 20 yen, cos you can also rephrase this '切る' as '割る'
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user1016May 19 '12 at 4:48

On Space ALC, the first relevant definition if you just search for 切る is probably: break《ゴルフ》（スコアがあるポイントを) But that only works if you know what "break 100" means in a golf context. Although you also see this used in other sporting contexts, e.g. "break the four minute mile".
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nkjtMay 19 '12 at 11:59

@DaveMG ""Cut to" and "cut from" are opposing concepts, so how can the same word do both?" Doesn't "cut" do them both in English? It does in your formulations.
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dainichiMay 22 '12 at 0:06

Don't think about "cut". Stop thinking about "cut from" or "cut down to" or anything else. At a certain point, thinking in English about Japanese becomes more obstructive than anything else. Just think in Japanese.

Well, OK, the example from your question (300万まん人にんを切きった。) isn't the most intuitive. Let's go with another example. I first encountered this meaning in the context of a race, and I think that's a great way to understand this meaning, so we'll go with that (lifting Chocolate's example from another comment...). Again, only think in Japanese.

This is a very informative answer... except that I'm not sure that "thinking in English" is really the problem here. I have a definition in my head for 切る, but it's not that it's an English definition, it's just the definition I have. I'm only using English as one way of expressing that definition here on this site, because I don't have the Japanese ability to express my definitions in Japanese. So, I guess what I'm saying is, don't mistake talking about Japanese words in English with defining Japanese words in English. Other than that, though, a helpful answer. +1 :)
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QuestionerMay 19 '12 at 0:16

@Dave M G: Well, the main idea (not explicitly stated in the answer) is, discard your preconceptions - in whatever language - as much as possible and just focus on what's in the sentence, and what doesn't make sense in that sentence, and how to make it make sense. I actually kind of think of this meaning of 切る like breaking a wall, perhaps with some sharp, decisive (cutting?) action.
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SomethingJapaneseMay 19 '12 at 1:17

I think I'm starting to see it now. 切る is just "cut", not "cut to" or "cut from". Whether not something is cut "to" or "from" comes from the grammar around it. Which is like English, now that I think of it. But, unlike English, the default assumption of whether to focus on the remainder or the source after the cutting is done might be opposite. In any case, to test if I've got this right, would it be grammatical to say 「1000メートル競走に10秒が切きった」 to mean I cut ten seconds off my time? Or would it have to be 「1000メートル競走に10秒から切きった」?
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QuestionerMay 20 '12 at 3:32

I'm not sure I actually see that meaning in 切る's definition. The JE entry includes "reduce/decrease/discount", but I'm not sure what Japanese definition that's supposed to fit, so I don't trust that it's the same as what you're trying to say here. I'd feel much more comfortable using 削{けず}る. Maybe a native speaker can give you a more confident answer.
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SomethingJapaneseMay 20 '12 at 4:09

EDIT it won't let me do to the previous...goo actually says 「元値を―・って売る」, or "discount" it seems, but I still don't know how to use that well enough to confidently tell you how to say what you want to (supposing you can).
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SomethingJapaneseMay 20 '12 at 4:16

I'm totally unfamiliar with this definition of 切る. It's also not coming up in my dictionaries. Is this a common usage?
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QuestionerMay 17 '12 at 6:19

Yes, I think it's quite common, and I think it's a bit more conversational than 下回る. My smaller Japanese dictionary (岩波国語辞典) says "切る②㋓《助詞「を」を介し、値段・量などの基準を示す語と合して》下回る。割る。「原価を切って売る」「10秒を切る」". One of my Japanese-English dictionaries says "切る--④[下回る]100メートル競走に10秒を切った He broke the 10 second mark in the 100-meter race. かばんを元値を切って売っている They are selling the bags below costs."
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user1016May 17 '12 at 7:34